Oriented Coloring of a Grid Abdullah Makkeh Tartu likool October - - PowerPoint PPT Presentation

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Oriented Coloring of a Grid

Abdullah Makkeh

Tartu Ülikool

October 4, 2015

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Co-author: Bahman Ghandchi

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

General View

1

Background

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

General View

1

Background

2

Oriented Chromatic number

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

General View

1

Background

2

Oriented Chromatic number

3

Oriented Chromatic number of a grid

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

General View

1

Background

2

Oriented Chromatic number

3

Oriented Chromatic number of a grid

4

Integer programming models

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Notations

If G is a graph, we denote by V (G) its set of vertices and by E(G) its set of edges.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Notations

If G is a graph, we denote by V (G) its set of vertices and by E(G) its set of edges. If G is a digraph, we denote by V (G) its set of vertices and by A(G) : {(x, y)|if the arc is form x to y} its set of arcs.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Notations

If G is a graph, we denote by V (G) its set of vertices and by E(G) its set of edges. If G is a digraph, we denote by V (G) its set of vertices and by A(G) : {(x, y)|if the arc is form x to y} its set of arcs. The order of an undirected graph, or a digraph, is the cardinality of its vertex set.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Notations

If G is a graph, we denote by V (G) its set of vertices and by E(G) its set of edges. If G is a digraph, we denote by V (G) its set of vertices and by A(G) : {(x, y)|if the arc is form x to y} its set of arcs. The order of an undirected graph, or a digraph, is the cardinality of its vertex set. All the graphs we consider are simple and have no loops.

Homomorphism

Let G and G ′ be two graphs. A homomorphism of G to G ′ is a mapping f : V (G) → V (H) that preserves the edges: f (x)f (y) ∈ E(G ′) whenever xy ∈ E(G).

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Notations

If G is a graph, we denote by V (G) its set of vertices and by E(G) its set of edges. If G is a digraph, we denote by V (G) its set of vertices and by A(G) : {(x, y)|if the arc is form x to y} its set of arcs. The order of an undirected graph, or a digraph, is the cardinality of its vertex set. All the graphs we consider are simple and have no loops.

Homomorphism

Let G and G ′ be two graphs. A homomorphism of G to G ′ is a mapping f : V (G) → V (H) that preserves the edges: f (x)f (y) ∈ E(G ′) whenever xy ∈ E(G). If D and D′ are two digraphs, a homomorphism of D to D′ is a mapping f : V (D) → V (D′) that preserves the arcs: (f (x), f (y)) ∈ E(D′) whenever (x, y) ∈ E(D).

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Graph Coloring

A (proper) k-colouring of a graph G is a partition of V (G) into k parts, called colour classes, such that no two adjacent vertices belong to the same colour class.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Graph Coloring

A (proper) k-colouring of a graph G is a partition of V (G) into k parts, called colour classes, such that no two adjacent vertices belong to the same colour class.

χ(G) is the chromatic number of G, defined as the smallest k such

that G admits a k-colouring.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Graph Coloring

A (proper) k-colouring of a graph G is a partition of V (G) into k parts, called colour classes, such that no two adjacent vertices belong to the same colour class.

χ(G) is the chromatic number of G, defined as the smallest k such

that G admits a k-colouring. Such a k-colouring can be equivalently regarded as a homomorphism

• f G to the complete graph Kk on k vertices.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Graph Coloring

A (proper) k-colouring of a graph G is a partition of V (G) into k parts, called colour classes, such that no two adjacent vertices belong to the same colour class.

χ(G) is the chromatic number of G, defined as the smallest k such

that G admits a k-colouring. Such a k-colouring can be equivalently regarded as a homomorphism

• f G to the complete graph Kk on k vertices.

χ(G) corresponds to the smallest k such that G admits a

homomorphism to Kk.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Graph Coloring

A (proper) k-colouring of a graph G is a partition of V (G) into k parts, called colour classes, such that no two adjacent vertices belong to the same colour class.

χ(G) is the chromatic number of G, defined as the smallest k such

that G admits a k-colouring. Such a k-colouring can be equivalently regarded as a homomorphism

• f G to the complete graph Kk on k vertices.

χ(G) corresponds to the smallest k such that G admits a

homomorphism to Kk. Is there other types of coloring on digraphs?

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Oriented coloring

An oriented k-colouring of a digraph D is a partition of V (D) into k colour classes such that:

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Oriented coloring

An oriented k-colouring of a digraph D is a partition of V (D) into k colour classes such that:

1

No two adjacent vertices belong to the same colour class.

2

All the arcs connecting every two colour classes have the same direction.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Oriented coloring

An oriented k-colouring of a digraph D is a partition of V (D) into k colour classes such that:

1

No two adjacent vertices belong to the same colour class.

2

All the arcs connecting every two colour classes have the same direction.

It is a mapping γ from V (D) to a set of k colours such that:

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Oriented coloring

An oriented k-colouring of a digraph D is a partition of V (D) into k colour classes such that:

1

No two adjacent vertices belong to the same colour class.

2

All the arcs connecting every two colour classes have the same direction.

It is a mapping γ from V (D) to a set of k colours such that:

1

γ(u) = γ(v) for every arc (u, v) in A(D).

2

γ(u) = γ(x) for every two arcs (u, v) and (w, x) with γ(v) = γ(w).

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Oriented coloring

An oriented k-colouring of a digraph D is a partition of V (D) into k colour classes such that:

1

No two adjacent vertices belong to the same colour class.

2

All the arcs connecting every two colour classes have the same direction.

It is a mapping γ from V (D) to a set of k colours such that:

1

γ(u) = γ(v) for every arc (u, v) in A(D).

2

γ(u) = γ(x) for every two arcs (u, v) and (w, x) with γ(v) = γ(w).

The oriented chromatic number χo(D) is the smallest k for which D admits an oriented k-colouring.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Oriented coloring

An oriented k-colouring of a digraph D is a partition of V (D) into k colour classes such that:

1

No two adjacent vertices belong to the same colour class.

2

All the arcs connecting every two colour classes have the same direction.

It is a mapping γ from V (D) to a set of k colours such that:

1

γ(u) = γ(v) for every arc (u, v) in A(D).

2

γ(u) = γ(x) for every two arcs (u, v) and (w, x) with γ(v) = γ(w).

The oriented chromatic number χo(D) is the smallest k for which D admits an oriented k-colouring. The notion of oriented chromatic number can be extended to graphs.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Oriented coloring

An oriented k-colouring of a digraph D is a partition of V (D) into k colour classes such that:

1

No two adjacent vertices belong to the same colour class.

2

All the arcs connecting every two colour classes have the same direction.

It is a mapping γ from V (D) to a set of k colours such that:

1

γ(u) = γ(v) for every arc (u, v) in A(D).

2

γ(u) = γ(x) for every two arcs (u, v) and (w, x) with γ(v) = γ(w).

The oriented chromatic number χo(D) is the smallest k for which D admits an oriented k-colouring. The notion of oriented chromatic number can be extended to graphs.

χo(G) is defined as the maximum of the oriented chromatic

numbers of its orientations.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Examples χo(C▽) is 5 where C5 is a cycle on five vertices.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Examples χo(C▽) is 5 where C5 is a cycle on five vertices. χ(Kk,k) = 2 where Kk,k is the complete bipartite digraph.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Examples χo(C▽) is 5 where C5 is a cycle on five vertices. χ(Kk,k) = 2 where Kk,k is the complete bipartite digraph.

But any two vertice in Kk,k are connected by directed path of length 2.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Examples χo(C▽) is 5 where C5 is a cycle on five vertices. χ(Kk,k) = 2 where Kk,k is the complete bipartite digraph.

But any two vertice in Kk,k are connected by directed path of length 2.

χo(Kk,k) = 2k.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Tournament Coloring

An oriented k-colouring of D can be equivalently regarded as a homomorphism of D to some oriented graph Hk of order k.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Tournament Coloring

An oriented k-colouring of D can be equivalently regarded as a homomorphism of D to some oriented graph Hk of order k. The vertices of Hk are then used as colours and such a homomorphism is called an Hk-colouring.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Tournament Coloring

An oriented k-colouring of D can be equivalently regarded as a homomorphism of D to some oriented graph Hk of order k. The vertices of Hk are then used as colours and such a homomorphism is called an Hk-colouring. When Hk is a tournament. Then the coloring is tournament coloring.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Tournament Coloring

An oriented k-colouring of D can be equivalently regarded as a homomorphism of D to some oriented graph Hk of order k. The vertices of Hk are then used as colours and such a homomorphism is called an Hk-colouring. When Hk is a tournament. Then the coloring is tournament coloring.

χo(D) corresponds to the minimum order of an oriented graph H

such that D admits a homomorphism to H.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Tournament Coloring

An oriented k-colouring of D can be equivalently regarded as a homomorphism of D to some oriented graph Hk of order k. The vertices of Hk are then used as colours and such a homomorphism is called an Hk-colouring. When Hk is a tournament. Then the coloring is tournament coloring.

χo(D) corresponds to the minimum order of an oriented graph H

such that D admits a homomorphism to H.

χo(G) is defined as the maximum of the oriented chromatic

numbers of its members, where G is a class of graphs.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Tournament Coloring

An oriented k-colouring of D can be equivalently regarded as a homomorphism of D to some oriented graph Hk of order k. The vertices of Hk are then used as colours and such a homomorphism is called an Hk-colouring. When Hk is a tournament. Then the coloring is tournament coloring.

χo(D) corresponds to the minimum order of an oriented graph H

such that D admits a homomorphism to H.

χo(G) is defined as the maximum of the oriented chromatic

numbers of its members, where G is a class of graphs.

Lemma

Let G be a class of graphs satisfying, H1, H2 ∈ G ∃ G ∈ G;G ⊃ H1, H2. Then ∃ T on max

G∈G χo(G) vertices ∀ G ∈ G.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Grids

A 2-dimensional grid Gm,n is an undirected graph with vertices V = {(i, j)|0 ≤ i ≤ m;0 ≤ j ≤ n} and edges of the form

{(i, j),(i +1, j)}or{(i, j),(i, j +1)}.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Example: Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Example: Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Trivial Bound on χo

A digraph D is an oriented clique (an o-clique) if χo(D) = |V (G)|.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Trivial Bound on χo

A digraph D is an oriented clique (an o-clique) if χo(D) = |V (G)|. A digraph is an o-clique iff every two of its vertices are connected by a directed path of length 1 or 2.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Trivial Bound on χo

A digraph D is an oriented clique (an o-clique) if χo(D) = |V (G)|. A digraph is an o-clique iff every two of its vertices are connected by a directed path of length 1 or 2. The absolute oriented clique number ao(D) is the maximum order of an o-clique subgraph of D.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Trivial Bound on χo

A digraph D is an oriented clique (an o-clique) if χo(D) = |V (G)|. A digraph is an o-clique iff every two of its vertices are connected by a directed path of length 1 or 2. The absolute oriented clique number ao(D) is the maximum order of an o-clique subgraph of D. The relative oriented clique number ro(D) is the maximum cardinality of a subset S of D such that every two vertices of S are connected in D by a directed path of length 1 or 2.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Trivial Bound on χo

A digraph D is an oriented clique (an o-clique) if χo(D) = |V (G)|. A digraph is an o-clique iff every two of its vertices are connected by a directed path of length 1 or 2. The absolute oriented clique number ao(D) is the maximum order of an o-clique subgraph of D. The relative oriented clique number ro(D) is the maximum cardinality of a subset S of D such that every two vertices of S are connected in D by a directed path of length 1 or 2.

ωao(D) ≤ ωro(D) ≤ χo(D).

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Theorem(Fertin, Raspaud and Roychowdhury [2])

For every integers m,n ≥ 1, χo(Gm,n) ≤ 11.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References proof: Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References proof:

Property P1:In T11 for any two vertices there exists at least two distinct paths of length 2 in K11, for any orientation of the path.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References proof:

Property P1:In T11 for any two vertices there exists at least two distinct paths of length 2 in K11, for any orientation of the path. (1st row Coloring) Color the first row by an homomorphism in T11, (the indegree and outdegree of any vertex in T11 is 5)

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References proof:

Property P1:In T11 for any two vertices there exists at least two distinct paths of length 2 in K11, for any orientation of the path. (1st row Coloring) Color the first row by an homomorphism in T11, (the indegree and outdegree of any vertex in T11 is 5) (2nd row Coloring)

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References proof:

Property P1:In T11 for any two vertices there exists at least two distinct paths of length 2 in K11, for any orientation of the path. (1st row Coloring) Color the first row by an homomorphism in T11, (the indegree and outdegree of any vertex in T11 is 5) (2nd row Coloring)

1

Let (0,1) and (1,0) have different colors(outdegree of any vertex in T11 is 5).

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References proof:

Property P1:In T11 for any two vertices there exists at least two distinct paths of length 2 in K11, for any orientation of the path. (1st row Coloring) Color the first row by an homomorphism in T11, (the indegree and outdegree of any vertex in T11 is 5) (2nd row Coloring)

1

Let (0,1) and (1,0) have different colors(outdegree of any vertex in T11 is 5).

2

By Property P1 (1,1) and (0,2) have different colors by an homomorphism in T11

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References proof:

Property P1:In T11 for any two vertices there exists at least two distinct paths of length 2 in K11, for any orientation of the path. (1st row Coloring) Color the first row by an homomorphism in T11, (the indegree and outdegree of any vertex in T11 is 5) (2nd row Coloring)

1

Let (0,1) and (1,0) have different colors(outdegree of any vertex in T11 is 5).

2

By Property P1 (1,1) and (0,2) have different colors by an homomorphism in T11

3

We can recursively continue the coloring of row 2.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References proof:

Property P1:In T11 for any two vertices there exists at least two distinct paths of length 2 in K11, for any orientation of the path. (1st row Coloring) Color the first row by an homomorphism in T11, (the indegree and outdegree of any vertex in T11 is 5) (2nd row Coloring)

1

Let (0,1) and (1,0) have different colors(outdegree of any vertex in T11 is 5).

2

By Property P1 (1,1) and (0,2) have different colors by an homomorphism in T11

3

We can recursively continue the coloring of row 2.

We can color row (r +1)th when row (r)th is colored by an homomorphism in T11.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Tournament k-colouring

Input: Gm,n(V ,E) k ∈ N Feasible Solution: Tournament colouring of Gm,n with Tk. Minimize\Maximize: Nothing! Just find a solution!

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

min 0 Subjected to:

∀v ∈ V :

• j∈[k]

xv,j = 1 [every vertex has exactly 1 color]

∀ (u, v) ∈ E(G) ∀ j ∈ [k] :

xu,j +xv,j ≤ 1 [adj. vertex has different color]

∀ (u, v) ∈ E(G)∀ i, j ∈ [k] :

xu,i +xv,j ≤ yi,j +1 [Tournament Colouring]

∀ (i, j) ∈ Tk :

yi,j binary

∀v ∈ V ∀ j ∈ [k]

xv,j binary

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

min 0 Subjected to:

∀v ∈ V :

• j∈[k]

xv,j = 1 [every vertex has exactly 1 color]

∀ (u, v) ∈ E(G) ∀ j ∈ [k] :

xu,j +xv,j ≤ 1 [adj. vertex has different color]

∀ (u, v) ∈ E(G)∀ i, j ∈ [k] :

xu,i +xv,j ≤ yi,j +1 [Tournament Colouring]

∀ (i, j) ∈ Tk :

yi,j binary

∀v ∈ V ∀ j ∈ [k]

xv,j binary This approach is very slow (!!!) Since it will run over all tournaments.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

min 0 Subjected to:

∀v ∈ V :

• j∈[k]

xv,j = 1 [every vertex has exactly 1 color]

∀ (u, v) ∈ E(G) ∀ j ∈ [k] :

xu,j +xv,j ≤ 1 [adj. vertex has different color]

∀ (u, v) ∈ E(G)∀ i, j ∈ [k] :

xu,i +xv,j ≤ yi,j +1 [Tournament Colouring]

∀ (i, j) ∈ Tk :

yi,j binary

∀v ∈ V ∀ j ∈ [k]

xv,j binary This approach is very slow (!!!) Since it will run over all tournaments. In next model we will run it over non-isometric tournaments only.

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Tournament k-colouring

Input: Gm,n(V ,E) k ∈ N A ∈ Mk(Z) Feasible Solution: Tournament colouring of Gm,n with Tk having A as the adjacency matrix. Minimize\Maximize: Nothing! Just find a solution!

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

min 0 Subjected to:

∀v ∈ V :

• j∈[k]

xv,j = 1 [every vertex has exactly 1 color]

∀ (u, v) ∈ E(G) ∀ j ∈ [k] :

xu,j +xv,j ≤ 1 [adj. vertex has different color]

∀ (u, v) ∈ E(G)∀ i, j ∈ [k] :

xu,i +xv,j ≤ yi,j +1 [Tournament Colouring]

∀ (i, j) ∈ Tk :

yi,j = ai,j [specifying Tk ]

∀v ∈ V ∀ j ∈ [k]

xv,j binary

Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Abdullah Makkeh Oriented Coloring of a Grid

Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References

Dybizbański, Janusz, and Anna Nenca. "Oriented chromatic number

• f grids is greater than 7." Information Processing Letters 112.4

(2012): 113-117. Fertin, Guillaume, André Raspaud, and Arup Roychowdhury. "On the oriented chromatic number of grids." Information Processing Letters 85.5 (2003): 261-266. Sopena, E. "Homomorphisms and colourings of oriented graphs: An updated survey." Discrete Mathematics (2015). Szepietowski, Andrzej, and Monika Targan. "A note on the oriented chromatic number of grids." Information processing letters 92.2 (2004): 65-70.

Abdullah Makkeh Oriented Coloring of a Grid